STAT217 Tutorial sheet #5

Question 1

In a steel plant, the hardness is supposed to be uniformly distributed between 65 and 75 on a certain scale. For a sample of 100 pieces, this was the distribution:

Hardness	65 to under 67.5	67.5 to under 70	70 to under 72.5	72.5 to under 75
Frequency	28	30	23	19

Is the distribution uniform? Test at a 5% level of significance. (c² = 2.96; conclude the distribution is uniform)

Question 2

A factory examines 7 widgets at a time for quality control purposes. It estimates that 24% of the widgets will be defective on average. For 500 samples, each consisting of 7 widgets, the factory had the following distribution of the number of defective widgets:

Value	0	1	2	3	4	5	6	7
Observations	85	159	144	76	32	4	0	0

Here is the appropriate binomial distribution:

X	0	1	2	3	4	5	6	7
P(x)	0.1465	0.3237	0.3067	0.1614	0.0510	0.0097	0.0010	0.0000

Is the factory estimate correct? Test at a 5% level of significance (c² = 4.7763; conclude the data follows a binomial distribution with n=7 and p=0.24)

Question 3

A city will install a traffic light at an intersection if it averages 3 vehicles per minute. In a sample of 500 minutes, the following distribution was observed:

Value	0	1	2	3	4	5	6	7	8	9
Observations	26	84	120	109	74	51	21	12	1	2

Here is the appropriate Poisson distribution:

X	0	1	2	3	4	5	6	7	8	9
P(x)	0.0498	0.1494	0.2240	0.2240	0.1680	0.1008	0.0504	0.0216	0.0081	0.0027

Does the data follow a Poisson distribution with a mean of 3? (c² = 5.3892; conclude the data follows a Poisson distribution with mean = 3)

Question 4

In a survey at a mall during August, people were asked how much they spent that day. For a sample of 8 people these were the results:

50

75

100

120

140

150

240

1350

a) Is the data normally distributed? Test at a 5% level of significance. You may use the following table:

Value	Z score	F(z)
50	-0.52	0.3015
75	-0.46	0.3228
100	-0.41	0.3409
120	-0.36	0.3594
140	-0.32	0.3745
150	-0.29	0.3859
240	-0.09	0.4641
1350	2.45	0.9929

(test stat = 0.4109; data is not normally distributed)

b) Clearly 1350 is an outlier. If we remove this value, show why the remaining data is normally distributed. You may use the following table:

Value	Z score	F(z)
50	-1.22	0.1112
75	-0.81	0.209
100	-0.41	0.3409
120	-0.08	0.4681
140	0.24	0.5948
150	0.41	0.6591
240	1.86	0.9686

(test statistic = 0.198; data is normally distributed)

Question 5

For a social service agency, this is the distribution of home status by gender:

	Male	Female
Stable home	3658	8099
Home with problems	138	293
Transient home situation	357	708
Homeless	2450	3047

a) Does a person’s home situation depend on gender? Test at a 5% level of significance. (c² = 300.8206; conclude home status depends on gender)

b) To what degree does home status depend on gender? (12.67%)